5.
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6.
7.
8.
9.
(A) 8 3
(B) 6 3
(C) 4 3
(D) 2 3
The circles on focal radii of a parabola as diameter touch:
(A) the tangent at the vertex
(B) the axis
(C) the directrix
(D) none of these
The equation of the tangent to the parabola y = (x − 3)2 parallel to the chord joining the points
(3, 0) and (4, 1) is:
(A) 2 x − 2 y + 6 = 0
(B) 2 y − 2 x + 6 = 0
(C) 4 y − 4 x + 11 = 0 (D) 4 x − 4 y = 11
The angle between the tangents drawn from a point ( – a, 2a) to y 2 = 4 ax is
π
π
π
π
(A)
(B)
(C)
(D)
3
6
4
2
An equation of a tangent common to the parabolas y2 = 4x and x 2 = 4y is
(A) x – y + 1 = 0
(B) x + y – 1 = 0
(C) x + y + 1 = 0
(D) y = 0
The line 4x − 7y + 10 = 0 intersects the parabola, y2 = 4x at the points A & B. The co-ordinates of the
point of intersection of the tangents drawn at the points A & B are:
 7 5
, 
 2 2
(A) 
7
 5
,− 
 2
2
(B)  −
 5 7
, 
 2 2
(C) 
5
 7
,− 
 2
2
(D)  −
AP & BP are tangents to the parabola, y 2 = 4x at A & B. If the chord AB passes through a fixed point
(− 1, 1) then the equation of locus of P is
(B) y = 2 (x + 1)
(C) y = 2 x
(D) y 2 = 2 (x − 1)
(A) y = 2 (x − 1)
11.
Equation of the normal to the parabola, y 2 = 4ax at its point (am 2, 2 am) is:
(A) y = − mx + 2am + am 3
(B) y = mx − 2am − am 3 (C) y = mx + 2am + am 3
(D) none
12.
At what point on the parabola y2 = 4x the normal makes equal angles with the axes?
(A) (4, 4)
(B) (9, 6)
(C) (4, – 1)
(D) (1, 2)
13.
If on a given base, a triangle be described such that the sum of the tangents of the base angles is a
constant, then the locus of the vertex is:
(A) a circle
(B) a parabola
(C) an ellipse
(D) a hyperbola
14.
A point moves such that the square of its distance from a straight line is equal to the difference
between the square of its distance from the centre of a circle and the square of the radius of the circle.
The locus of the point is:
(A)
a straight line at right angles to the given line (B)
a circle concentric with the given circle
(C)a parabola with its axis parallel to the given line(D) a parabola with its axis perpendicular to the given line.
15.
P is any point on the parabola, y 2 = 4ax whose vertex is A. PA is produced to meet the directrix in D &
M is the foot of the perpendicular from P on the directrix. The angle subtended by MD at the focus is:
(A) π/4
(B) π/3
(C) 5π/12
(D) π/2
16.
If the distances of two points P & Q from the focus of a parabola y2 = 4ax are 4 & 9, then the distance
of the point of intersection of tangents at P & Q from the focus is:
(A) 8
(B) 6
(C) 5
(D) 13
17.
Tangents are drawn from the point (− 1, 2) on the parabola y2 = 4 x. The length of intercept made by
these tangents on the line x = 2 is:
10.
18.
19.
20.
21.
22.
23.
(A) 6
(B) 6 2
(C) 2 6
(D) none of these
From the point (4, 6) a pair of tangent lines are drawn to the parabola, y2 = 8x. The area of the triangle
formed by these pair of tangent lines & the chord of contact of the point (4, 6) is:
(A) 8
(B) 4
(C) 2
(D) none of these
Locus of the intersection of the tangents at the ends of the normal chords of the parabola
y 2 = 4ax is
(A)(2a + x) y 2 + 4a3 = 0
(B) (2a + x) + y2 = 0
(C) (2a + x) y2 + 4a = 0
(D) none of these
If the tangents & normals at the extremities of a focal chord of a parabola intersect at
(x 1, y1) and (x 2, y2) respectively, then:
(B) x 1 = y2
(C) y1 = y 2
(D) x 2 = y 1
(A) x 1 = x 2
Tangents are drawn from the points on the line x − y + 3 = 0 to parabola y 2 = 8x. Then all the chords of
contact passes through a fixed point whose coordinates are:
(A) (3, 2)
(B) (2, 4)
(C) (3, 4)
(D) (4, 1)
The distance between a tangent to the parabola y2 = 4 A x (A > 0) and the parallel normal with gradient
1 is:
(A) 4 A
(B) 2 2 A
(C) 2 A
(D) 2 A
A variable parabola of latus ractum , touches a fixed equal parabola, then axes of the two curves being
60 of 91 CONIC SECTION
98930 58881 , BHOPAL, (M.P.)
Part : (A) Only one correct option
1.
If (2, 0) is the vertex & y − axis the directrix of a parabola, then its focus is:
(A) (2, 0)
(B) (− 2, 0)
(C) (4, 0)
(D) (− 4, 0)
2.
A parabol a is drawn wi th i t s f ocus at (3, 4) and v ert ex at t he f ocus of t he parabol a
y 2 − 12 x − 4 y + 4 = 0. The equation of the parabola is:
(A) x 2 − 6 x − 8 y + 25 = 0
(B) y2 − 8 x − 6 y + 25 = 0
(C) x 2 − 6 x + 8 y − 25 = 0
(D) x 2 + 6 x − 8 y − 25 = 0
3.
The length of the chord of the parabola, y2 = 12x passing through the vertex & making an angle of 60º
with the axis of x is:
(A) 8
(B) 4
(C) 16/3
(D) none
4.
The length of the side of an equilateral triangle inscribed in the parabola, y2 = 4x so that one of its
angular point is at the vertex is:
TEKO CLASSES, H.O.D. MATHS : SUHAG R. KARIYA (S. R. K. Sir) PH: (0755)- 32 00 000,
EXERCISE–10
26.
27.
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28.
29.
30.
(B)
a3
p2
(C)
4a3
p2
32.
33.
35.
36.
π
, then the locus of P is:
4
(A) x − y + 1 = 0
(B) x + y − 1 = 0
(C) x − y − 1 = 0
(D) x + y + 1 = 0
Locus of the point of intersection of the normals at the ends of parallel chords of gradient m of the
parabola y2 = 4ax is:
(A) 2 xm 2 − ym 3 = 4a (2 + m 2) (B) 2 xm 2 + ym 3 = 4a (2 + m 2)
(C) 2 xm + ym 2 = 4a (2 + m)
(D) 2 xm 2 − ym 3 = 4a (2 − m 2)
The equation of the other normal to the parabola y 2 = 4ax which passes through the intersection of
those at (4a, − 4a) & (9a, − 6a) is:
(A) 5x − y + 115 a = 0 (B) 5x + y − 135 a = 0 (C) 5x − y − 115 a = 0 (D) 5x + y + 115 = 0
The point(s) on the parabola y2 = 4x which are closest to the circle,
x 2 + y 2 − 24y + 128 = 0 is/are:
(
34.
p2
a
AB is a chord of the parabola y2 = 4ax with vertex at A. BC is drawn perpendicular to AB meeting the
axis at C. The projection of BC on the axis of the parabola is
(A) a
(B) 2a
(C) 4a
(D) 8a
The locus of the foot of the perpendiculars drawn from the vertex on a variable tangent to the parabola
y 2 = 4ax is:
(B) y (x 2 + y2) + ax 2 = 0
(A) x (x 2 + y2) + ay2 = 0
(C) x (x 2 − y2) + ay 2 = 0
(D) none of these
T is a point on the tangent to a parabola y 2 = 4ax at its point P. TL and TN are the perpendiculars on the
focal radius SP and the directrix of the parabola respectively. Then:
(A) SL = 2 (TN)
(B) 3 (SL) = 2 (TN)
(C) SL = TN
(D) 2 (SL) = 3 (TN)
The point of contact of the tangent to the parabola y 2 = 9x which passes through the point
(4, 10) and makes an angle θ with the axis of the parabola such that tan θ > 2 is
(A) (4/9, 2)
(B) (36, 18)
(C) (4, 6)
(D) (1/4, 3/2)
If the parabolas y2 = 4x and x2 = 32 y intersect at (16, 8) at an angle θ, then θ is equal to
3
4
π
(A) tan–1  
(B) tan–1  
(C) π
(D)
5
5
 
 
2
From an external point P, pair of tangent lines are drawn to the parabola, y 2 = 4x. If θ1 & θ2 are the
inclinations of these tangents with the axis of x such that, θ1 + θ2 =
31.
(D)
)
(C) (4, 4)
(D) none
(A) (0, 0)
(B) 2 , 2 2
If P1 Q1 and P2 Q2 are two focal chords of the parabola y2 = 4ax, then the chords P1P2 and Q1Q2 intersect on
the
(A) directrix
(B) axis
(C) tangent at the vertex (D) none of these
If x + y = k, is the normal to y2 = 12x, then k is
[IIT - 2000]
(A) 3
(B) 9
(C) –9
(D) – 3
The equation of the common tangent touching the circle (x – 3)2 + y2 = 9 and the parabola y2 = 4x above the
x-axis is
[IIT - 2001]
(A)
3 y = 3x + 1
(B)
3 y = –(x + 3)
(C)
3 y =x + 3
(D)
3 y = –(3x + 1)
37.
The focal chord to y 2 = 16 x is tangent to (x − 6)2 + y2 = 2, then the possible values of the slope of this
chord are:
[IIT - 2003]
(A) {− 1, 1}
(B) {− 2, 2}
(C) {− 2, 1/2}
(D) {2, − 1/2}
38.
The normal drawn at a point (at12, –2at1) of the parabola y2 = 4ax meets it again in the point (at22, 2at2), then
[IIT - 2003]
2
2
2
2
(A) t2 = t1 + t
(B) t2 = t1 –
(C) t2 = –t1 +
(D) t2 – t1 –
t1
t1
t1
1
39.
The angle between the tangents drawn from the point (1, 4) to the parabola y2 = 4x is
40.
π
π
π
π
(A)
(B)
(C)
(D)
3
6
2
4
Let P be the point (1, 0) and Q a point of the locus y2 = 8x. The locus of mid point of PQ is
[IIT - 2004]
(A) x 2 + 4y + 2 = 0
41.
(B) x 2 – 4y + 2 = 0
(C) y2 – 4x + 2 = 0
[IIT - 2005]
(D) y2 + 4x + 2 = 0
A parabola has its vertex and focus in the first quadrant and axis along the line y = x. If the distances
of the vertex and focus from the origin are respectively 2 and 2 2 , then an equation of the parabola
is
[IIT - 2006]
(A) (x + y)2 = x – y + 2
(B) (x – y)2 = x + y – 2
(C) (x – y)2 = 8(x + y – 2)
(D) (x + y) 2 = 8(x – y + 2)
61 of 91 CONIC SECTION
25.
2a2
p
98930 58881 , BHOPAL, (M.P.)
(A)
TEKO CLASSES, H.O.D. MATHS : SUHAG R. KARIYA (S. R. K. Sir) PH: (0755)- 32 00 000,
24.
parallel. The locus of the vertex of the moving curve is a parabola, whole latus rectum is:
(A) (B) 2 (C) 4 (D) none
Length of the focal chord of the parabola y2 = 4ax at a distance p from the vertex is:
42.
43.
44.
Part
45.
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46.
47.
48.
If P is a point on C1 and Q in another point on C2,
is equal to
[IIT - 2006 ]
QA 2 + QB 2 + QC 2 + QD 2
(A) 0.75
(B) 1.25
(C) 1
(D) 0.5
A circle touch the line L and the circle C1 externally such that both the circles are on the same side of the
line, then the locus of centre of the circle is
[IIT - 2006 ]
(A) ellipse
(B) hyperbola
(C) parabola
(D) parts of straight line
A line M through A is drawn parallel to BD. Point S moves such that its distances from the line BD and the
vertex A are equal. If locus of S cuts M at T2 and T3 and AC at T1, then area of ∆T1T2T3 is [IIT - 2006)]
2
1
(A)
sq. units
(B)
sq. units
(C) 1 sq. units
(D) 2 sq. units
3
2
: (B) May have more than one options correct
If one end of a focal chord of the parabola y2 = 4x is (1, 2), the other end lies on
(B) xy + 2 = 0
(C) xy – 2 = 0
(D) x 2 + xy – y – 1 = 0
(A) x 2 y + 2 = 0
The tangents at the extremities of a focal chord of a parabola
(A) are perpendicular
(B) are parallel
(C) intersect on the directrix
(D) intersect at the vertex
If from a variable point 'P' pair of perpendicular tangents PA and PB are drawn to any parabola then
(A)
P lies on directrix of parabola
(B)
chord of contact AB passes through focus
(C)
chord of contact AB passes through of fixed point
(D)
P lies on director circle
A normal chord of the parabola subtending a right angle at the vertex makes an acute angle θ with the
x − axis, then θ =
(A) arc tan 2
49.
50.
51.
PA 2 + PB 2 + PC 2 + PD 2
(B) arc sec 3
π
(C) arc cot 2
(D)
− arc cot 2
2
Variable chords of the parabola y2 = 4ax subtend a right angle at the vertex. Then:
(A)
locus of the feet of the perpendiculars from the vertex on these chords is a circle
(B)
locus of the middle points of the chords is a parabola
(C)
variable chords passes through a fixed point on the axis of the parabola (D) none of these
Two parabolas have the same focus. If their directrices are the x − axis & the y − axis respectively, then
the slope of their common chord is:
(A) 1
(B) − 1
(C) 4/3
(D) 3/4
P is a point on the parabola y2 = 4ax (a > 0) whose vertex is A. PA is produced to meet the directrix in
D and M is the foot of the perpendicular from P on the directrix. If a circle is described on MD as a
diameter then it intersects the x−axis at a point whose co−ordinates are:
(A) (− 3a, 0)
(B) (− a, 0)
(C) (− 2a, 0)
(D) (a, 0)
EXERCISE–11
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
Find the vertex, axis, focus, directrix, latusrectum of the parabola x 2 + 2y – 3x + 5 = 0.
Find the set of values of α in the interval [π/2, 3π/2], for which the point (sinα, cosα) does not lie outside the
parabola 2y2 + x – 2 = 0.
Two perpendicular chords are drawn from the origin ‘O’ to the parabola y = x2, which meet the parabola at P
and Q Rectangle POQR is completed. Find the locus of vertex R.
Find the equation of tangent & normal at the ends of the latus rectum of the parabola
y 2 = 4a (x – a).
Prove that the straight line x + my + n = 0 touches the parabola y2 = 4ax if n = am 2.
If tangent at P and Q to the parabola y2 = 4ax intersect at R then prove that mid point of R and M lies
on the parabola, where M is the mid point of P and Q.
Find the equation of normal to the parabola x2 = 4y at (9, 6).
Find the equation of the chord of y2 = 8x which is bisected at (2, – 3)
Find the locus of the mid-points of the chords of the parabola y2 = 4ax which subtend a right angle at the
vertex of the parabola.
Find the equation of the circle which passes through the focus of the parabola x 2 = 4 y & touches it at
the point (6, 9).
Prove that the normals at the points, where the straight line x + my = 1 meets the parabola y 2 = 4ax,
 4am2 4am 
 of the parabola.
meet on the normal at the point  2 ,

 If the normals at three points P, Q, and R on parabola y2 = 4ax meet in a point O and S be the focus, prove
that SP. SQ . SR = a. SO2.
Show that the locus of the point of intersection of the tangents to y2 = 4ax which intercept a constant length
d on the directrix is (y2 – 4ax) (x + a)2 = d2 x 2.
Show that the distance between a tangent to the parabola y2 = 4ax and the parallel normal is
a sec2θ cosec θ, where θ is the inclination of the either with the axis of the parabola.
P and Q are the point of contact of the tangents drawn from a point R to the parabola y2 = 4ax. If PQ be a
normal to the parabola at P, prove that PR is bisected by the directrix.
A circle is described whose centre is the vertex and whose diameter is three-quarters of the latus
rectum of the parabola y2 = 4ax. If PQ is the common chord of the circle and the parabola and L1 L2 is
62 of 91 CONIC SECTION
Let ABCD be a square of side length 2 units. C2 is the circle through vertices A, B, C, D and C1 is the circle
touching all the sides of the square ABCD. L is a line through A.
98930 58881 , BHOPAL, (M.P.)
[IIT - 2006]
TEKO CLASSES, H.O.D. MATHS : SUHAG R. KARIYA (S. R. K. Sir) PH: (0755)- 32 00 000,
Comprehension
19.
2
described on the focal distance of the given point as diameter is a 1+ t .
A parabola is drawn to pass through A and B, the ends of a diameter of a given circle of radius a, and
to have as directrix a tangent to a concentric circle of radius b; then axes being AB and a perpendicular
diameter, prove that the locus of the focus of the parabola is
20.
FREE Download Study Package from website: www.tekoclasses.com
21.
x2
+
y2
=1
b2
b2 − a2
PNP′ is a double ordinate of the parabola then prove that the locus of the point of intersection of the
norm al at P and the st raight line through P′ paral lel to the axis is the equal parabola
y 2 = 4a (x – 4a).
Find the locus of the point of intersection of those normals to the parabola x 2 = 8 y which are at right
angles to each other.
[IIT - 1997]
22.
Let C1 and C2 be respectively, the parabolas x 2 = y – 1 and y2 = x – 1. Let P be any point on C1 and Q
be any point on C2. Let P1 and Q 1 be the reflections of P and Q, respectively, with respect to the line
y = x. Prove that P1 lies on C2, Q 1 lies on C1 and PQ ≥ min {PP1 , QQ 1}. Hence or otherwise determine
points P0 and Q 0 on the parabolas C1 and C2 respectively such that P0 Q 0 ≤ PQ for all pairs of points
(P, Q) with P on C1 and Q on C2.
[IIT - 2000]
23.
Normals are drawn from the point P with slopes m1, m 2, m 3 to the parabola y2 = 4x. If locus of P with
m 2 m 2 = α is a part of the parabola itself then find α.
[IIT - 2003]
EXERCISE–10
EXERCISE–11
1. C
2. A
3. A
4. A
5. A
6. D
7. B
8. C
9. C
10. A
11. A
12. D
13. B
14. D
15. D
16. B
17. B
18. C
19. A
20. C
21. C
axis x = 3, directrix y = –
22. B
23. B
24. C
25. C
26. A
27. C
28. A
2. α ∈ [π/2, 5π/6] ∪ [π, 3π/2]
29. A
30. C
31. A
32. B
33. C
34. A
35. B
4. Tangent y = x, y = – x,
Normal x + y = 4a, x – y = 4a
36. C
37. A
38. A
39. B
40. C
41. C
42. A
7. 2x + 9y = 72
43. C
44. C
45. ABD
48. BD 49. ABC
50. AB
46. AC 47. ABCD
51. AD
29 
3
 , focus
1. vertex ≡  , −
2
8 

33 
3
 ,−

2
8 

29
. Latus rectum = 2.
3
3. y2 = x – 2
8. 4x + 3y + 1 = 0
9. y2 – 2ax + 8a2 = 0
10. x 2 + y 2 + 18x – 28y + 27 = 0
21. x 2 − 2 y + 12 = 0
23. α = 2
63 of 91 CONIC SECTION
18.
If the normals from any point to the parabola x 2 = 4y cuts the line y = 2 in points whose abscissa are
in A.P., then prove that slopes of the tangents at the 3 conormal points are in GP.
Prove that the length of the intercept on the normal at the point (at 2, 2at) made by the circle which is
98930 58881 , BHOPAL, (M.P.)
17.
TEKO CLASSES, H.O.D. MATHS : SUHAG R. KARIYA (S. R. K. Sir) PH: (0755)- 32 00 000,
 2 + 2
 a2.
 2 
the latus rectum, then prove that the area of the trapezium PL1 L2Q is 